Problem: Factor the following expression: $-5$ $x^2+$ $26$ $x$ $-24$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(-24)} &=& 120 \\ {a} + {b} &=& & & {26} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $120$ and add them together. The factors that add up to ${26}$ will be your ${a}$ and ${b}$ When ${a}$ is ${6}$ and ${b}$ is ${20}$ $ \begin{eqnarray} {ab} &=& ({6})({20}) &=& 120 \\ {a} + {b} &=& {6} + {20} &=& 26 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-5}x^2 +{6}x +{20}x {-24} $ Group the terms so that there is a common factor in each group: $ ({-5}x^2 +{6}x) + ({20}x {-24}) $ Factor out the common factors: $ x(-5x + 6) - 4(-5x + 6) $ Notice how $(-5x + 6)$ has become a common factor. Factor this out to find the answer. $(-5x + 6)(x - 4)$